Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities

TL;DR

This paper generalizes the formula for the volume of a hypercube clipped by multiple hyperplanes, extending Lawrence's formula and deriving explicit volume formulas and combinatorial identities.

Contribution

It introduces a generalized volume formula for hypercubes intersected by multiple hyperplanes, relaxing previous restrictions and deriving new combinatorial identities.

Findings

Explicit volume formulas for clipped hypercubes

Generalization of Lawrence's formula

New combinatorial identities

Abstract

There is an elegant expression for the volume of hypercube $[0,1]^n$ clipped by a single hyperplane. In the article the formula is generalized to the case of more than one hyperplane. An important foundation for the result is Lawrence's formula and a way to weaken two restrictions of simplicity and non-parallelness in his formula is also considered. Several concrete volume formulas of clipped hypercubes are derived explicitly and the corresponding combinatorial identities are obtained as an application.